I have heard its possible to find the sine and cosine of some common angles without calculator.

Can anyone explain how please?

eg.

cos pi/4 = 1/√2
sin pi/4 = 1/√2

cos 5pi/4 = -1/√2

cos pi/3 = 1/2

sin pi/3 = √3/2

Thanks in advance

Nov 1st 2012, 06:29 AM

BobP

Re: Finding the sine, cos of common angles without calculator

Look at the thread when tanx = 1

Nov 1st 2012, 07:06 AM

Soroban

Re: Finding the sine, cos of common angles without calculator

Hello, ScorpFire!

Quote:

I have heard its possible to find the sine and cosine of some common angles without calculator.

Can anyone explain how please?

For example: .

, consider an isosceles right triangle.

Code:

*
* *
* * h
1 * *
* *
* 45 *
* * * * *
1

Let the equal sides equal 1.
Pythagorus says the hypotenuse is

We have: .
And you can write the trig values of

Memorize "one, one, square-root-of-two".

, consider an equilateral triangle. . with side length 2. .Draw an altitude.

Code:

*
*|*
* | *
2 * |y *
* | *
* 60 | *
* * * * *
: 1 : 1 :

We have: .
Pythagorus says:
And you can write the trig values for

For , turn the above right triangle on its side.

Code:

*
2 * *
* * 1
* 30 *
* * *_ * *
√3

We have: .
And you can write the trig values for

For both diagrams, memorize "one, two, square-root-of-three".

Be careful! .Remember that:. . The shortest side, 1, is opposite the smallest angle, 30o.. . The longest side, 2, is opposite the largest angle, 90o.

Nov 1st 2012, 07:50 AM

richard1234

Re: Finding the sine, cos of common angles without calculator

Take a point on the unit circle. The cosine of the corresponding angle is simply the x-coordinate (adjacent over hypotenuse). The sine of that angle is the y-coordinate. That's how I usually find sine/cosine in my head by just visualizing the unit circle.

Nov 2nd 2012, 10:14 PM

Salahuddin559

Re: Finding the sine, cos of common angles without calculator

There are formulae for finding sin(A+B) and cos(A+B) etc... using these one can extend the values for other angles. For example, sin2A = 2sinAcosA, using which one can find sin(15 degrees) = sqrt(2 - sqrt(3))/2.